110 PROCEEDINGS OF THE AMERICAN ACADEMY. 



suggest that the function a(t) has a negligible effect upon the motion. 

 It therefore follows that (4) or the equation 



_ + jft. - „ _ = o, (5) 



obtained therefrom by discarding the factor dx/dt, is at best only 

 approximate. Concerning the degree of approximation we may 

 perhaps be willing to admit that the error is always extremely small ; 

 but we should be very chary about admitting that the error is neces- 

 sarily small relative to the terms retained. 



That (5) is only an approximate result is clearly brought out by 

 Planck in his derivation, but the clearness is somewhat obscured by 

 a footnote referring to Abraham's derivation in a manner which 

 implies, though it does not state, that the latter proof is exact. Abra- 

 ham gives in the first instance essentially the proof which we have 

 given above. He goes on, however, to show by a discussion of 

 the physical dimensions of the terms that the equation (5) is the 

 only possible one. What he really proves is that under certain hy- 

 potheses (5) is the only possible linear equation of the form 



(px ii» v div 



d¥ +kx= ^^W 



The physical dimensions are equally well satisfied by the non-linear 

 equation 



dx 

 It 



— 4- /' 2 7- 



df + A X 



12 ,\ 2 



( m) = °' (6) 



which follows naturally and immediately from (2), that is, from the 

 principle of energy, provided it is assumed that the rate of radiation 

 is proportional at each instant to the square of the acceleration, instead 

 of merely proportional thereto when integrated over a complete 

 period, and which would arise from the equation in integrated form, 

 provided one were willing to apply to it directly the reasoning which 

 is applied after an integration by parts. 



Lorentz has also derived (5) by a method which throws emphasis 

 on the approximate nature of the result. 3 That he regards the equa- 

 tion with some suspicion may be inferred from his statement of ;i 

 later date 4 : Eine Gleichung "die iibrigens den Mathematikern noch 

 manche Frage darbote." 



3 See, for example, his Theory of Electrons, 251, (1909). 



4 Physikalische Zeitschrift, 11, 1250 (1910) It was this remark which so 

 strengthened my own suspicions as to lead me to take up the question some 

 time ago. 



